Question

Two positive numbers have their HCF as 12 and their product as 6336. Choose the correct option for the number of pairs possible for the given numbers.
A. 2
B. 3
C. 4
D. 5

Answer 1

Hint: Here we assume two numbers as different variables and use the concept of HCF that HCF of two numbers is the highest common factor of two numbers, which implies that HCF must divide the two numbers. We write the two numbers in the form of multiples of HCF. Take the product of the numbers and equate it to the value given in the question to get the possible pairs.
* The HCF of two numbers is the highest common factor that divides both the numbers.
* If a divided b then we can write \[b = m \times a\]where m is another factor of b.

Complete step-by-step answer:
Let us assume the two numbers as x and y.
Then HCF of two numbers x and y is 12.
Since, we know HCF of two numbers is the highest common factor that divides both the numbers.
12 divides x\[ \Rightarrow \]\[x = 12m\], for any value of m
12 divides y\[ \Rightarrow \]\[y = 12n\], for any value of n
So the two numbers are \[x = 12m,y = 12n\]
Now we know the product of two numbers is 6336.
\[ \Rightarrow x \times y = 6336\]
Substitute the values of x and y
\[ \Rightarrow (12m) \times (12n) = 6336\]
\[ \Rightarrow 144 \times m \times n = 6336\]
Divide both sides by 144
\[ \Rightarrow \dfrac{{144 \times m \times n}}{{144}} = \dfrac{{6336}}{{144}}\]
Cancel the same terms from numerator and denominator.
\[ \Rightarrow m \times n = 44\]
Now we factorize the RHS i.e.\[44 = 11 \times 4 \times 1\]
So, possible pairs of numbers can be:
44,1 ; 1,44 ; 11,4 ; 4,11
So, there are 4 pairs.

So, the correct option is C.

Note: Students might make mistakes when they don’t understand what the statement a divides b means and end up writing \[a = bn\] for any number n, which is wrong. Keep in mind we keep the value of that element on one side which is being divided and the element that divides the number becomes a factor or a multiple of the number.